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A numerical method for coupled surface and grain boundary motion

Published online by Cambridge University Press:  01 June 2008

ZHENGUO PAN  and
BRIAN WETTON
ZHENGUO PAN
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C. CanadaV6T 1Z email: [email protected]
BRIAN WETTON
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C. CanadaV6T 1Z2 email: [email protected]
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Abstract

We study the coupled surface and grain boundary motion in a bi-crystal in the context of the ‘quarter loop’ geometry. Two types of normal curve velocities are involved in this model: motion by mean curvature and motion by surface diffusion. Three curves meet at points where junction conditions are given. A formulation that describes the coupled normal motion of the curves and preserves arc length parametrisation up to scaling is proposed. The formulation is shown to be well-posed in a simple, linear setting. Equations and junction conditions are approximated by finite difference methods. Numerical convergence to exact travelling wave solutions is shown. The method is applied to other problems of physical interest.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 3 , June 2008 , pp. 311 - 327
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Ascher, U. M. & Petzold, L. R. (1998) Computer methods for ordinary differential equations and differential-algebraic equations. SIAM. Philadelphia, PA.CrossRefGoogle Scholar
[2]Bansch, E., Morin, P. & Nochetto, R. H. (2005) A finite element method for surface diffusion: The parametric case. J. Comput. Phys. 203 (1), 321–343.CrossRefGoogle Scholar
[3]Barrett, J. W., Garckeb, H. & Nurnberg, R. (2007) On the variational approximation of combined second and fourth order geometric evolution equations. SIAM J. Sci. Comput. 29 (3), 1006–1041.CrossRefGoogle Scholar
[4]Barrett, J. W., Garckeb, H. & Nurnberg, R. (2007) A parametric finite element method for fourth order geometric evolution equations. J. Comput. Phys. 222, 441–467.CrossRefGoogle Scholar
[5]Brokman, A., Kris, R., Mullins, W. W. & Vilenkin, A. J. (1995) Analysis of boundary motion in thin films. Scr. Metall. Mater. 32, 1341–1346.CrossRefGoogle Scholar
[6]Bronsard, L. & Reitich, F. (1993) On three-phase boundary motion and the singular limit of a vector-valued Ginzburg–Landau equation. Arch. Ration. Mech. Anal. 124 (4), 355–379.CrossRefGoogle Scholar
[7]Bronsard, L. & Wetton, B. T. R. (1995) A numerical method for tracking curve networks moving with curvature motion. J. Comput. Phys. 120, 66–87.CrossRefGoogle Scholar
[8]Cahn, J. & Taylor, J. (1994) Overview 113: Surface motion by surface diffusion. Acta Metall. Mater. 42, 1045–1063.CrossRefGoogle Scholar
[9]David, L. C. & Sethian, J. A. (1999) Motion by intrinsic Laplacian of curvature. Interfaces Free Boundarie. 1, 1–18.Google Scholar
[10]Deckelnick, K., Dziuk, G. & Elliott, C. M. (2005) Computation of geometric partial differential equations and mean curvature flow. Acta Numeric. 14, 139–232.CrossRefGoogle Scholar
[11]Dunn, C. G. & Sharp, M. (1952) Secondary recrystallization texture in Copper. Trans AIM. 194, 42–43.Google Scholar
[12]Dziuk, G., Kuwert, E. & Schatzle, R. (2002) Evolution of elastic curves in Rn: Existence and computation. SIAM J. Math. Anal. 33, 1228–1245.CrossRefGoogle Scholar
[13]Garcke, H. & Novick-Cohen, A. (2000) A singular limit for a system of degenerate cahn-hilliard equations. Adv. Differ. Equs. 5, 401–434.Google Scholar
[14]Hou, T. Y., Lowengrub, J. S. & Shelley, M. J. (2001) Boundary integral methods for multicomponent fluids and multiphase materials. JC. 169, 302–362.Google Scholar
[15]Kanel, J. & Novick-Cohen, A. (2004) Coupled surface and grain boundary motion: Nonclassical traveling wave solutions. Adv. Differ. Equs. 9, 299–327.Google Scholar
[16]Kanel, J., Novick-Cohen, A. & Vilenkin, A. (2003) A traveling wave solution for coupled surface and grain boundary motion. Acta Mater. 51, 1981–1989.CrossRefGoogle Scholar
[17]Kanel, J., Novick-Cohen, A. & Vilenkin, A. (2004) Coupled surface and grain boundary motion: A travelling wave solution. Nonlinear Anal. 59, 1267–1292.CrossRefGoogle Scholar
[18]Kanel, J., Novick-Cohen, A. & Vilenkin, A. (2005) A numerical study of grain boundary motion in bicrystals. Acta Mater. 53, 227–235.CrossRefGoogle Scholar
[19]Min, D. & Wong, H. (2004) A model of migrating grain-boundary grooves with application to two mobility-measurement methods. Acta Mater. 50, 5155–5169.CrossRefGoogle Scholar
[20]Mikula, K. & Sevcovic, D. (2004) Computational and qualitative aspects of curves driven by curvature and external force. Comput. Vis. Sci. 6, 211–225.CrossRefGoogle Scholar
[21]Mullins, W. W. (1957) Theory of thermal grooving. J. Appl. Phys. 28, 333–339.CrossRefGoogle Scholar
[22]Pan, Z. & Wetton, B. Numerical methods for coupled surface and grain boundary motion, arXiv report math.NA/0702503. URL: http://arxiv.org/. Submitted 2007, last updated February 16, 2007.Google Scholar
[23]Ruuth, S. J. (1998) Efficient algorithms for diffusion-generated motion by mean curvature. J. Comput. Phys. 144, 603–625.CrossRefGoogle Scholar
[24]Esedoglu, S. & Tsai, R., Threshold dynamics for high order geometric motions. To appear in Interfaces Free Boundaries.Google Scholar
[25]Smereka, P. (2003) Semi-Implicit level set methods for curvature and surface diffusion motion. J. Sci. Comput. 19(1–3), 439–456.CrossRefGoogle Scholar
[26]Solonnikov, V. A. (1965) Boundary value problems in physics, In: Proceedings of the Steklov Institute of Mathematics, Vol. 83, pp. 487–491.Google Scholar
[27]Sun, B., Suo, Z. & Yang, W. (1997) A finite element method for simulating interface motion, part I: Migration of phase and grain boundaries. Acta Mater. 45, 1907–1915.CrossRefGoogle Scholar
[28]Sun, B. & Suo, Z. (1997) A finite element method for simulating interface motion, part II: Large shape changes due to surface diffusion. Acta Mater. 45, 4953–4962.CrossRefGoogle Scholar
[29]Suo, Z. (1997) Motions of microscopic surfaces in materials. Adv. Appl. Mech. 33, 193–294.CrossRefGoogle Scholar
[30]Vilenkin, A. J. & Kris, R. (1997) Breakup and grain growth in thin-film array. J. Appl. Phys. 81, 238–245.CrossRefGoogle Scholar
[31]Zhang, H. & Wong, H. (2002) Coupled grooving and migration of inclined grain boundaries: Regime I. Acta Mater. 50, 1983–1994.CrossRefGoogle Scholar
[32]Zhang, H. & Wong, H. (2002) Coupled grooving and migration of inclined grain boundaries: Regime II. Acta Mater. 50, 1995–2012.CrossRefGoogle Scholar
[33]Zhang, W. & Schneibel, J. H. (1995) Numerical Simulation of grain-boundary grooving by surface diffusion. Comput. Mater. Sci. 3, 347–358.CrossRefGoogle Scholar